Wash chicken quarters and dry using a clean cloth or paper towel. Prick chicken all over with a sharp kitchen knife and place in a bowl. Mix lime juice, ginger, garlic, cream, cardamom, cashew nut paste, salt, pepper, and oil together in a medium bowl. Rub chicken all over with lime mixture. Transfer to a baking dish or large shallow bowl and marinate for 2 to 4 hours in the refrigerator.
Preheat a grill to medium. Grill chicken quarters for about 25 minutes or until fully cooked, turning pieces over halfway through cooking.